You will be dealing with more elements, more branches, more nodes, more loops and more brain-breaking problems. But all of these may be lessen turmoil may be lighten if you have understood very well the concepts of the Ohm’s Law and Kirchoff’s Current and Voltage laws(KCL and KVL) from the previous entries prior to this one.
Expected that you are now prepared to apply these laws to develop two powerful techniques for circuit analysis namely:
Nodal Analysis which is based on a systematic application of Kirchoff’s Current Law (KCL) and
Mesh Analysis which is based on a systematic application of Kirchoff’s Voltage Law (KVL). These two techniques are too significant and the students must not miss to dig deeper. Therefore, students are encouraged to pay much careful attention to this topic.
First analysis to be tackled is the Nodal Analysis.
NODAL ANALYSIS WITHOUT VOLTAGE SOURCE
This analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables. It is more convenient to choose node voltages instead of element voltages and it reduces the number of equations that must solve simultaneously. MAKES YOU NOT TOO CRAZY!!!In nodal analysis, we are interested in finding the node voltages. Given a circuit with n nodes without voltage sources, here are the steps to take to determine Node Voltages.
- Select a node as the reference node. Assign voltages v1, v2, … vn-1, to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.
- Apply KCL to each of the n-1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages. (It is n-1 because you have to subtract 1 which refers to the reference node which has zero V already. So you just have to focus for the remaining nodes)
- Solve the resulting simultaneous equations to obtain the unknown node voltages.
Let us now talk furthermore about each step.
ON FIRST STEP:
Selecting reference Node or datum node is the first step in this analysis. The reference node is commonly called the ground and it is assumed to have a zero voltage. Commonly, we use the earth ground (a) or (b).
Keep in mind that Node 0 is reference node(v=0), while nodes 1 and 2 are assigned voltages v1 and v2. Always bear that node voltages are defined with respect to the reference node.
ON SECOND STEP:
Apply KCL to each nonreference node in the circuit. We advised you to redraw the circuits and put labels on each node with current and voltage labels. This is to avoid putting too much information on the same circuits and this will ease your CRAZINESS!!!
After this, sum up all currents i1, i2, i3 through their corresponding resistors R1, R2, R3,respectively.
At node 1, applying KCL gives I1=I2+i1 + i2
At node 2, I1+I2 = i3
We can now apply Ohm’s Law to express the unknown currents i1, i2 and i3 in terms of nodes voltages.
The key idea to bear in mind is that, since resistance is a passive element, by passive sign convention, current must always flow from a higher potential to a lower potential.
*Current flows from a higher potential to a lower potential in a resistor.
To be precised, this principle is set to be
I= vhigher - vlower
R
- i1= v1 – 0
- i2= v1 – v2
- I3= v2 – 0
Substitute all of the equations a, b, and c to the equations of node 1 and node 2.
After that, you have to simplify for you to take it easy for using Cramer’s Rule. (You are probably 3rd year or 4th year now, so we expect you to have knowledge about Cramer’s Rule)
Get the determinant first of the matrix coming from the equations on node 1 and node 2. (delta)
After getting the determinant, divide the delta1 from the delta. So as for the delta2 to get the v1 and v2.
NODAL ANALYSIS WITH VOLTAGE SOURCE
Let's consider that the voltage can affect the nodal analysis. Using this circuit, we can have two possibilities.
CASE 1
If a voltage source is connected between the reference node and a nonreference node, we simple set the voltage at the nonreference node equal to the voltage of the voltage source. In the figure, for example,
V1= 10 V
Thus , our analysis is somewhat simplified by this knowledge of the voltage at this node.
CASE 2
If the voltage source(dependent or independent) is connected between two nonreference nodes form a generalized node or supernode, we both now apply KCL and KVL to determine the node voltages.
where a supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.
In this figure, nodes 2 and 3 form a supernode. We analyze a circuits with supernodes using the same three steps from behind but supernodes are just treated the other way on this matter. This is because KCL is essential component of nodal analysis which required knowing the current through each element. But there’s no way to know the current through a voltage source ahead. So, KCL must be satisfied at a supernode.
i1 + i4= i2 + i3
But how can you fill the i’s?
Remember the I= vhigher - vlower
R
You may apply it here. So,
v1 – v2 + v1 – v2 = v2 – 0 + v3 –0
4 2 8 6
To apply Kirchoff’s voltage law to the supernode in the figure, we redraw the circuits and make a loop in clockwise direction. This will give you
-v2 + 5 + v3 = 0 to v2 - v3 = 5
We now obtain the node voltages. Note the following properties of a supernode:
- The voltage source inside the supernode provides constraint equation needed to solve for the node voltages.
- A supernode has no voltage of its own.
- A supernode requires the application of both KCL and KVL.
Example:
LEARNING/SUMMARY
- Here we go now this one of the complicated topics that we have encountered. We do not want you to suffer like how we had it. You are now dealing with your confused mind. The best tip we can offer is to really familiarize all the topics prior to this one because this is also one of the applications of the basics.
- Maybe at first you get a hard time, but practicing exercises are very useful.
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